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| [[Image:Doubling oriented.svg|300px|right|thumb|A Doubling-oriented Doche-Icart-Kohel curve of equation <math>y^2=x^3-x^2-16x</math>]]
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| In [[mathematics]], the '''doubling-oriented Doche–Icart–Kohel curve''' is a form in which an [[elliptic curve]] can be written. It is a special case of [[Weierstrass form]] and it is also important in [[elliptic curve cryptography|elliptic-curve cryptography]] because the doubling speeds up considerably (computing as composition of 2-[[isogeny]] and its [[dual abelian variety|dual]]).
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| It has been introduced by Christophe Doche, Thomas Icart, and David R. Kohel in <ref>Christophe Doche, Thomas Icart, and David R. Kohel, ''Efficient Scalar Multiplication by Isogeny Decompositions''</ref>
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| ==Definition==
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| Let <math> K </math> be a [[field (mathematics)|field]] and let <math>a\in K</math>. Then, the Doubling-oriented Doche–Icart–Kohel curve with [[parameter]] ''a'' in [[affine space|affine coordinates]] is represented by:
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| <math> y^2=x^3+ax^2+16ax </math> | |
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| Equivalently, in [[projective space|projective coordinates]]:
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| <math> ZY^2=X^3+aZX^2+16aXZ^2, </math>
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| with <math> x=\frac{X}{Z} </math> and <math>y=\frac{Y}{Z} </math>.
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| Notice that, since this curve is a special case of [[elliptic curve|Weierstrass form]], transformations to the most common form of elliptic curve (Weierstrass form) are not needed.
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| ==Group law==
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| It is interesting to analyze the [[elliptic curve#The group law|group law]] in [[elliptic curve cryptography]], defining the addition and doubling formulas, because these formulas are necessary to compute multiples of points ''[n]P'' (see [[Exponentiation by squaring]]). In general, the group law is defined in the following way: if three points lies in the same line then they sum up to zero. So, by this property, the group laws are different for every curve shape.
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| In this case, since these curves are special cases of Weierstrass curves, the addition is just the standard addition on Weierstrass curves. On the other hand, to double a point, the standard doubling formula can be used, but it would not be so fast.
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| In this case, the [[identity element|neutral element]] is <math> \theta=(0:1:0) </math> (in projective coordinates), for which <math> \theta=-\theta </math>. Then, if <math>P=(x,y)</math> is a non-trivial element (<math>P!=O</math>), then the inverse of this point (by addition) is –P=(x,-y).
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| ===Addition===
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| In this case, [[affine space|affine coordinates]] will be used to define the addition formula:
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| (x<sub>1</sub>,y<sub>1</sub>)+(x<sub>2</sub>,y<sub>2</sub>)=(x<sub>3</sub>,y<sub>3</sub>) where
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| x<sub>3</sub> = (-x<sub>1</sub><sup>3</sup>+(x<sub>2</sub>-a)x<sub>1</sub><sup>2</sup>+(x<sub>2</sub><sup>2</sup>+2ax<sub>2</sub>)x<sub>1</sub>+(y<sub>1</sub><sup>2</sup>-2y<sub>2</sub>y<sub>1</sub>+(-x<sub>2</sub><sup>3</sup>-ax<sub>2</sub><sup>2</sup>+y<sub>2</sub><sup>2</sup>)))/(x<sub>1</sub><sup>2</sup>-2x<sub>2</sub>x<sub>1</sub>+x<sub>2</sub><sup>2</sup>)
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| y<sub>3</sub> = ((-y<sub>1</sub>+2y<sub>2</sub>)x<sub>1</sub><sup>3</sup>+(-ay<sub>1</sub>+(-3y<sub>2</sub>x<sub>2</sub>+ay<sub>2</sub>))x<sub>1</sub><sup>2</sup>+((3x<sub>2</sub><sup>2</sup>+2ax<sub>2</sub>)y<sub>1</sub>-2ay<sub>2</sub>x<sub>2</sub>)x<sub>1</sub>+(y<sub>1</sub><sup>3</sup>-3y<sub>2</sub>y<sub>1</sub><sup>2</sup>+(-2x<sub>2</sub><sup>3</sup>-ax<sub>2</sub><sup>2</sup>+3y<sub>2</sub><sup>2</sup>)y<sub>1</sub>+(y<sub>2</sub>x<sub>2</sub><sup>3</sup>+ay<sub>2</sub>x<sub>2</sub><sup>2</sup>-y<sub>2</sub><sup>3</sup>)))/(-x<sub>1</sub><sup>3</sup>+3x<sub>2</sub>x<sub>1</sub><sup>2</sup>-3x<sub>2</sub><sup>2</sup>x<sub>1</sub>+x<sub>2</sub><sup>3</sup>)
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| ===Doubling===
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| 2(x<sub>1</sub>,y<sub>1</sub>)=(x<sub>3</sub>,y<sub>3</sub>)
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| x<sub>3</sub> = 1/(4y<sub>1</sub><sup>2</sup>)x<sub>1</sub><sup>4</sup>-8a/y<sub>1</sub><sup>2</sup>x<sub>1</sub><sup>2</sup>+64a2/y<sub>1</sub><sup>2</sup>
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| y<sub>3</sub> = 1/(8y<sub>1</sub><sup>3</sup>)x<sub>1</sub><sup>6</sup>+((-a<sup>2</sup>+40a)/(4y<sub>1</sub><sup>3</sup>))x<sub>1</sub><sup>4</sup>+((ay<sub>1</sub><sup>2</sup>+(16a<sup>3</sup>-640a<sup>2</sup>))/(4y<sub>1</sub><sup>3</sup>))x<sub>1</sub><sup>2</sup>+((-4a<sup>2</sup>y<sub>1</sub><sup>2</sup>-512a<sup>3</sup>)/y<sub>1</sub><sup>3</sup>)
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| ==Algorithms and examples==
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| ===Addition===
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| The fastest addition is the following one (comparing with the results given in: http://hyperelliptic.org/EFD/g1p/index.html), and the cost that it takes is 4 multiplications, 4 squaring and 10 addition.
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| A = Y<sub>2</sub>-Y<sub>1</sub>
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| AA = A<sup>2</sup>
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| B = X<sub>2</sub>-X<sub>1</sub>
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| CC = B<sup>2</sup>
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| F = X<sub>1</sub>CC
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| Z<sub>3</sub> = 2CC
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| D = X<sub>2</sub>Z<sub>3</sub>
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| ZZ<sub>3</sub> = Z<sub>3</sub><sup>2</sup>
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| X<sub>3</sub> = 2(AA-F)-aZ<sub>3</sub>-D
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| Y<sub>3</sub> = ((A+B)<sup>2</sup>-AA-CC)(D-X<sub>3</sub>)-Y<sub>2</sub>ZZ<sub>3</sub>
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| ====Example====
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| Let <math> K=\mathbb{Q} </math>. Let P=(X<sub>1</sub>,Y<sub>1</sub>)=(2,1), Q=(X<sub>2</sub>,Y<sub>2</sub>)=(1,-1) and a=1, then
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| A=2
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| AA=4
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| B=1
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| CC=1
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| F=2
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| ''' Z<sub>3</sub>=4'''
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| D=4
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| ZZ<sub>3</sub>=16
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| ''' X<sub>3</sub>=-4'''
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| '''Y<sub>3</sub>=336 '''
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| Thus, P+Q=(-4:336:4)
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| ===Doubling===
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| The following algorithm is the fastest one (see the following link to compare: http://hyperelliptic.org/EFD/g1p/index.html), and the cost that it takes is 1 multiplication, 5 squaring and 7 additions.
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| A = X<sub>1</sub><sup>2</sup>
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| B = A-a16
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| C = a<sub>2</sub>A
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| YY = Y<sub>1</sub><sup>2</sup>
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| YY<sub>2</sub> = 2YY
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| Z<sub>3</sub> = 2YY<sub>2</sub>
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| X<sub>3</sub> = B<sup>2</sup>
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| V = (Y<sub>1</sub>+B)2-YY-X<sub>3</sub>
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| Y<sub>3</sub> = V(X<sub>3</sub>+64C+a(YY<sub>2</sub>-C))
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| ZZ<sub>3</sub> = Z<sub>3</sub><sup>2</sup>
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| ====Example====
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| Let <math> K=\mathbb{Q} </math> and a=1. Let P=(-1,2), then Q=[2]P=(x3,y3) is given by:
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| A=1
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| B=-15
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| C=2
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| YY=4
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| YY<sub>2</sub>=8
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| '''Z<sub>3</sub>=16'''
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| '''X<sub>3</sub>=225'''
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| V=27
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| '''Y<sub>3</sub>=9693'''
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| ZZ<sub>3</sub>=256
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| Thus, Q=(225:9693:16).
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| ==Extended coordinates==
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| The addition and doubling computations should be as fast as possible, so it is more convenient to use the following representation of the coordinates:
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| <math> x,y </math> are represented by <math> X,Y,Z,ZZ </math> satisfying the following equations:
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| <math> x=\frac{X}{Z} </math>
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| <math> y=\frac{Y}{ZZ} </math>
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| <math> ZZ=Z^2 </math>
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| Then, the Doubling-oriented Doche–Icart–Kohel curve is given by the following equation:
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| <math> Y^2=ZX^3+aZ^2X^2+16aZ^3X </math>.
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| In this case,<math> P=(X: Y: Z: ZZ)</math> is a general point with inverse <math>-P=(X: -Y: Z: ZZ) </math>.
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| Furthermore, the points over the curve satisfy: <math> (X:Y:Z:Z^2)=(\lambda X: \lambda^2Y: \lambda Z: \lambda^2Z^2) </math> for all <math> \lambda </math> nonzero.
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| Faster doubling formulas for these curves and mixed-addition formulas were introduced by Doche, Icart and Kohel; but nowadays, these formulas are improved by Daniel J. Bernstein and Tanja Lange (see below the link of EFD).
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| ==Internal Link==
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| For more informations about the running-time required in a specific case, see [[Table of costs of operations in elliptic curves]]
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| ==External links==
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| * http://hyperelliptic.org/EFD/g1p/index.html
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite book
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| | author = Christophe Doche, Thomas Icart and David R. Kohel
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| | year = 2006
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| | title = Efficient Scalar Multiplication by Isogeny Decompositions
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| | publisher =Springer Berlin / Heidelberg
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| | url = http://www.springerlink.com/content/h542176232q8w45q/fulltext.pdf
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| | isbn = 978-3-540-33851-2
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| }}
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| * {{cite book
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| | author = Daniel J. Bernstein and Tanja Lange
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| | year = 2008
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| | title = Analysis and optimization of elliptic-curve single scalar multiplication
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| | publisher =
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| | url = http://books.google.nl/books?hl=es&lr=&id=VZ5kFYzH_ZUC&oi=fnd&pg=PA1&dq=related:0lMQ2OY5ejoJ:scholar.google.com/&ots=7erHZiu8CS&sig=PJCJlQQIhvqu0njfTyaV4DLYlkg#v=onepage&q=&f=false
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| | isbn =
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| }}
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| * http://www.hyperelliptic.org/EFD/g1p/auto-2dik.html
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| {{DEFAULTSORT:Doubling-oriented Doche-Icart-Kohel curve}}
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| [[Category:Elliptic curves]]
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| [[Category:Elliptic curve cryptography]]
| |
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